The Variational Principle

There has been very much effort devoted to the solution of the diffusion equation of motion for a reactant particle executing Brownian motion. The Euler equation of diffusion 

Much interest has developed on approximate techniques of solving quantum mechanical problems because exact solutions of the Schrodinger equation can not be obtained for many-body problems. One of the most convenient of such approximations for the solution of many-body problems is the application of the variational method. For instance, with approximate eigen-functions p , the eigen-values of the Hamiltonian H are En 

And the best estimate of the eigen-functions ipn is obtained by minimising the variational integral, and is given by E . This well-known procedure can be extended from the Schrodinger equation to the diffusion equation, since, in effect, both are diffusion equations [499]. 

So far, relatively little attention has been given to the variational method of solving diffusion problems. Nevertheless, it is a technique which may become of more interest as the nature of problems becomes more complex. Indeed, the variational method is the basis of the finite element method of numerical calculations and so is, in many ways, an equal alternative to the more familiar Crank—Nicholson approach [505a, 505b]. The author hopes that the comments made in this chapter will indicate how useful and versatile this approach can be. 

The power of quantum theory, as expressed in Eq. (4.1), is that if one has a molecular wave function in hand, one can calculate physical observables by application of the appropriate operator in a manner analogous to that shown for the Hamiltonian in Eq. (4.8). Regrettably, none of these equations offers us a prescription for obtaining the orthonormal set of molecular wave functions. Let us assume for the moment, however, that we can pick an arbitrary function,, which is indeed a function of the appropriate electronic and nuclear coordinates to be operated upon by the Hamiltonian. Since we defined the set of orthonormal wave functions 4, to be complete (and perhaps infinite), the function must be some linear combination of the 4 ,, i.e., 

let us consider evaluating the energy associated with wave function T. Taking the approach of multiplying on the left and integrating as outlined above, we have 

Assuming tlie coefficients to be real numbers, each tenn c must be greater than or equal to zero. By definition of Eq, the quantity ( , — Eq) must also be greater than or equal to zero. Thus, we have 

The variational principle asserts that any wavefunction constructed as a linear combination of orthonormal functions will have its energy greater than or equal to the lowest energy (E ) of the system. Thus, 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function  

In other words, the energy of the exact wavefunction serves as a lower bound to the energies calculated by any other normalized antisymmetric function. Thus, the problem becomes one of finding the set of coefficients that minimize the energy of the resultant wavefunction. 

Exploring Chemistry with Electronic Structure Methods 

To solve the problem of finding the best coefficients for the combinations in equations 3.41 and 3.46 one must accept here another non-demonstrated truth, called the variational principle. It states that whenever the true wavefunction is approximated by some incomplete function that depends on a number of parameters, the expectation value of the energy, equation 3.43, is higher than the expectation value that competes to the exact wavefunction. Briefly, whenever the wavefunction is wrong, the energy goes up. 

Prepare a molecular model with the positions of all nuclei, kept fixed because of the Born-Oppenheimer approximation. 

Choose a set of atomic orbitals xi. for each atom in the molecule, from repertoires available in the literature. These orbitals are given as a list of exponents in equation 3.40 and coefficients in equation 3.41. 

Guess a tentative set of coefficients cy, for example by diagonalizing the overlap matrix alone, and prepare tentative molecular orbitals. Symmetry plays a role here, because any molecular orbital must be either entirely symmetric or entirely antisymmetric with respect to symmetry elements in the molecule (remember what was said about normal coordinates in Section 2.1). 

Solve the secular equation and obtain a new set of coefficients (eigenvectors) and molecular orbital energies i (eigenvalues), and a new value of the total electronic energy. 

Nevertheless, the situation is not completely hopeless. There is a recipe for systematically approaching the wave function of the ground state XP0, i. e., the state which delivers the lowest energy E0. This is the variational principle, which holds a very prominent place in all quantum-chemical applications. We recall from standard quantum mechanics that the expectation value of a particular observable represented by the appropriate operator O using any, possibly complex, wave function f tnal that is normalized according to equation (1-10) is given by 

The variational principle now states that the energy computed via equation (1-11) as the expectation value of the Hamilton operator H from ary guessed vl tria will be an upper bound to the tme energy of the ground state, i. e., 

For example, f(x) = x2 + 1. Then, for x = 2, the function delivers y = 5. On the other hand, a functional needs a function as input, but again delivers a number  

Expectation values such as (O) in equation (1-11) are obviously functionals, since the value of (O) depends on the function Ftrial inserted. 

Coming back to the variational principle, the strategy for finding the ground state energy and wave function should be clear by now we need to minimize the functional Ej ] by searching through all acceptable N-electron wave functions. Acceptable means in this context that the trial functions must fulfill certain requirements which ensure that these func- 

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In the quantum mechanics of atoms and molecules, both perturbation theory and the variational principle are widely used. For some problems, one of the two classes of approach is clearly best suited to the task, and is thus an established choice. Flowever, in many others, the situation is less clear cut, and calculations can be done with either of the methods or a combination of both. 

Another connnon approximation is to construct a specific fonn for the many-body waveftmction. If one can obtain an accurate estimate for the wavefiinction, then, via the variational principle, a more accurate estimate for the energy will emerge. The most difficult part of this exercise is to use physical intuition to define a trial wavefiinction. 

When the reciprocal relations are valid in accord with (A3.2.251 then R is also symmetric. The variational principle in this case may be stated as... 

C. Lanczos, The Variational Principles of Mechanics , University of Toronto Press, Toronto, 1970... 

Using the variation principle to optimize Cj and C2 we obtain dE/dc and dEjdc from Equation (7.44) and put them equal to zero, giving... 

The solvated Fock operator can be naturally derived from the variational principles [14] defining the Helmlioltz free energy of the system (fA) by... 

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

The variational principle leads to the following equations describing the molecular orbital expansion coefficients, c. , derived by Roothaan and by Hall ... 

In order to test the accuracy of the LCAO approximations, we use the variation principle if V lcao is an approximate solution then the variational integral... 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

We now need to use the variation principle to seek the best possible values of the LCAO coefficients. To do this, I have to find Se as above, and set its first derivative to zero. I keep track of the requirement that the LCAO orbitals are... 

Then there is the question of quality. The variation principle only tells us about energies we might calculate the variational integral... 

It is also a common experience that traditional Cl calculations converge very poorly, because the virtual orbitals produced from an HF (or HF-LCAO) calculation are not determined by the variation principle and turn out to be very poor for representations of excited states. 

Sir William Hartree developed ingenious ways of solving the radial equation, and they are documented in Douglas R. Hartree s book (1957). By the time this book was published, the SCF method had been well developed, and its connection with the variation principle was finally understood. It is interesting to note that Chapter 2 of Douglas R. Hartree s book deals with the variation principle. 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The weight is the sum of coefficients at the given excitation level, eq. (4.2). The Cl method determines the coefficients from the variational principle, thus Table 4.2 shows that the doubly excited determinants are by far the most important in terms of energy. The singly excited determinants are the second most important, then follow the quadruples and triples. Excitations higher than 4 make only very small contributions, although there are actually many more of these highly excited determinants than the triples and quadruples, as illustrated in Table 4,1. 

The dissociation problem is solved in the case of a full Cl wave function. As seen from eq. (4.19), the ionic term can be made to disappear by setting ai = —no- The full Cl wave function generates the lowest possible energy (within the limitations of the chosen basis set) at all distances, with the optimum weights of the HF and doubly excited determinants determined by the variational principle. In the general case of a polyatomic molecule and a large basis set, correct dissociation of all bonds can be achieved if the Cl wave function contains all determinants generated by a full Cl in the valence orbital space. The latter corresponds to a full Cl if a minimum basis is employed, but is much smaller than a full Cl if an extended basis is used. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

Consider now making the variational coefficients in front of the inner basis functions constant, i.e. they are no longer parameters to be determined by the variational principle. The Is-orbital is thus described by a fixed linear combination of say six basis functions. Similarly the remaining four basis functions may be contracted into only two functions, for example by fixing the coefficient in front of the inner three functions. In doing this the number of basis functions to be handled by the variational procedure has been reduced from 10 to three. 

By including the doubly excited determinant, built from the antibonding MO, the amount of covalent and ionic terms may be varied, and be determined completely by the variational principle (eq. (4.19)). 

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